Let P (3 sec θ, 2 tan θ) and Q (3 sec φ, 2 tan φ) where θ + φ = (π/2), be two distinct points on the hyperbola (x2/9) - (y2/4) = 1. Then the ordinate of the point of intersection of the normals at P and Q is :

Question

Let P (3 sec θ, 2 tan θ) and Q (3 sec φ, 2 tan φ) where θ + φ = (π/2), be two distinct points on the hyperbola (x2/9) - (y2/4) = 1. Then the ordinate of the point of intersection of the normals at P and Q is : (A) (11/3) (B) (-11/3) (C) (13/2) (D) (-13/2)

Tardigrade Admin · Accepted Answer

p (3 sec θ, 2 tan θ) Q = (3 sec φ , 2 tanφ) θ +φ = (π/2) Q = (3 cosec θ , 2 cotθ) Equation of normal at p = = 3x cos θ + 2y cot θ=13 = 3x sin θ cos θ + 2y cos θ = 13 sinθ ...(1) equation of normal at Q ⇒ = 3x sin θ + 2y tan θ = 13 = 3x sin θ cos θ + 2y sin θ = 13 cosθ ...(2) (1)-(2) ⇒ 2y (cos θ - sin θ ) = 13 (sin θ - cos θ ) 2y = -13 ⇒ (-13/2)

Q. Let P $(3 sec θ, 2 t an θ)$ and Q $(3 sec ϕ, 2 t an ϕ)$ where $θ + ϕ = \frac{π}{2},$ be two distinct points on the hyperbola $\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1.$ Then the ordinate of the point of intersection of the normals at P and Q is :

$\frac{11}{3}$

$\frac{- 11}{3}$

$\frac{13}{2}$

$\frac{- 13}{2}$

Q. Let P (3secθ,2tanθ) and Q (3secϕ,2tanϕ) where θ+ϕ=2π​, be two distinct points on the hyperbola 9x2​−4y2​=1. Then the ordinate of the point of intersection of the normals at P and Q is :

311​

3−11​

213​

2−13​

Q. Let P $(3 sec θ, 2 t an θ)$ and Q $(3 sec ϕ, 2 t an ϕ)$ where $θ + ϕ = \frac{π}{2},$ be two distinct points on the hyperbola $\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1.$ Then the ordinate of the point of intersection of the normals at P and Q is :

$\frac{11}{3}$

$\frac{- 11}{3}$

$\frac{13}{2}$

$\frac{- 13}{2}$